Changes in the annual cycle of heavy precipitation across the British Isles within the 21st century

We model the annual cycle of heavy precipitation as an inhomogeneous Poisson process with a non-stationary threshold, for details see Coles (2001). Therefore we assume that the intensities of rare, heavy precipitation events can be described by the generalized Pareto distribution (Davison and Smith 1990) and that their occurrences are Poisson distributed and independent (to adapt the model to dependent data, the standard errors of the maximum likelihood estimates are adjusted (Fawcett and Walshaw 2007, and references therein)). To easily include the annual cycle via sinusoidal models for the location and the scale parameters the model has been reparametrized in terms of the generalized extreme value (GEV) distribution. This model has been shown to describe the annual cycle of heavy precipitation properly throughout the United Kingdom (Rust et al 2009, Maraun et al 2009). A short description of the statistical model, the choice of the threshold and the goodness-of-fit criterion is provided in section A.1, for details on the methods see Schindler et al (2012). The result of the goodness-of-fit analysis is given in the supplementary online material (SOM; available at stacks.iop.org/ERL/7/044029/mmedia).

The annual cycle is defined by the monthly return levels, z_p(m), for each month m of the year, defined as the value which is exceeded in month m with a probability p, e.g., z_0.04(Jan) (see section A.2). This corresponds to the 25 yr monthly return levels, which we will focus our analysis of the annual cycle on. The average over all 12 monthly return levels is denoted by $\overline{{z}_{p}(m)}$, defining an annual return level.

We define the peak time of the annual cycle of the monthly return levels as the month for which z_p(m) is maximal, i.e., argmax(z_p(m)). The amplitude is defined as half the range of the monthly return levels, z_p(m). Since this value is only meaningful in relation to the average monthly return level, we define the relative amplitude as the ratio of the amplitude and the average monthly return level, i.e., $1/2\ast (\max ({z}_{p}(m))-\min ({z}_{p}(m)))/\overline{{z}_{p}(m)}$. This relative value is comparable over regions with different magnitudes of monthly return levels.

All relevant characteristics of the annual cycle are summarized by the annual return level, peak time and relative amplitude. As annually and seasonally resolved changes of extreme precipitation across the UK are already well studied (Ekström et al 2005, Fowler and Ekström 2009, Fowler et al 2010, Buonomo et al 2007), we focus on peak times and relative amplitudes only. Schindler et al (2012) demonstrated that peak times are well represented by RCMs, but relative amplitudes are misrepresented for many regions. This insufficiency has to be kept in mind for the interpretation of any results related to the relative amplitudes.

In the situation of a very low relative amplitude, the peak time is not very robust and therefore also its change is not easily interpretable. The direction of change is defined as the sign of future minus reference peak times (relative amplitudes). A shift of the relative amplitude by more than 5% of the reference period's relative amplitude is considered a change.

The multi-model mean of the simulated peak times of the reference time period (1961–2000) is plotted in figure 1. It shows the typical east–west gradient with earlier peak times in the east and later peak times along the west coast of Great Britain and Ireland. For the two future time periods the twelve climate simulations vary considerably in the projection of the peak times (see SOM). In the multi-model mean for both future time periods, earlier or non-changing peak times are projected for the north-western regions and later peak times for the south-eastern regions (figure 2). The biggest shift is almost two months later for the far future (figure 2, right). This south-west–north-east division is also visible in figure 3, showing the fraction of climate simulations which project the same direction of change. For the far future, the signal is relatively strong in the south-eastern region. More than half of the projections agree on a shift of the peak times of the annual cycle towards later times in the year (figure 3, right). This signal is disturbed by internal climate fluctuations and hardly emerges for the mid future (figure 3, left). The change in the peak times is mostly consistent over different time periods: most climate simulations project a smaller or no change to the mid future and a bigger (or equal) change in the same direction for the last 40 years of the 21st century (figure 3 and SOM).

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The difference between eastern and western regions becomes gradually less pronounced (smaller difference and lesser spatial extent) in the two future projections (see SOM). There is no clear contribution of the driving AOGCM or the downscaling RCM to the change signal (not shown). However, there is a tendency that the driving AOGCM dominates the pattern of the direction of change, while no systematic influence of the RCM becomes apparent (see SOM).

The multi-model mean pattern of the relative amplitude over the reference period shows three distinct regions with high values (figure 4): at the west coast of Great Britain, in the west of Ireland and in eastern England, the monthly return levels vary with the season by at least 17% of the annual return level³ (up to 30% in Scotland). This pattern is similar to the one of the relative amplitude based on gridded observations (Schindler et al 2012). The direction of change depends strongly on the driving AOGCM (figure 5). ECHAM5-driven simulations project increasing relative amplitudes almost everywhere for both time periods (figure 5, left). HadCM3-driven simulations project decreases in southern and central England and increases for the far future everywhere else (figure 5, middle). A closer look at 31 arbitrary grid points⁴ located in England and Scotland (divided into four subregions) reveals remarkable differences (figure 6): for all grid points, ECHAM5-driven simulations show an upward trend, while ARPEGE- and BCM-driven simulations show mostly a negative signal; HadCM3-driven runs show greater variability and no common signal, even when subdivided by climate sensitivities. This behaviour depends on the driving AOGCM and to a lesser extent on the region (figure 6). Therefore also the common signal for the western regions (figure 5) might rather be due to internal climate fluctuations than due to anthropogenic forcing.

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The change in peak times for the southern and eastern regions of the UK towards later peak times is coherent with findings for observations by Fowler and Kilsby (2003b). They proposed changes in the Scandinavian pattern in autumn and North-Atlantic Oscillation (NAO) in winter as sources of the observed changes in peak times from summer to autumn⁵. However in the present RCM ensemble we do not find evidence that the Scandinavian pattern causes the change in peak times for eastern England: the Scandinavian pattern is well represented in the CMIP3 simulations (Handorf and Dethloff 2009), but there is no consistent response of the Scandinavian pattern to increased greenhouse gas emissions between global circulation models (Haugen and Iversen 2008). The projected change in peak times from summer to autumn is consistent across the simulations studied. Therefore the Scandinavian pattern cannot explain the consistent shift in the modelled peak time patterns.

The pattern-based NAO index shows an increase (for both summer and winter) in most AOGCMs considered in this study⁶ but the magnitudes of the increase spreads considerably (Bladé et al 2012a, Karpechko 2010, Handorf and Dethloff 2009). The winter NAO primarily influences heavy precipitation in western regions of the British Isles (Maraun et al 2012, Haylock and Goodess 2004, Scaife et al 2008, Santos et al 2007). Thus the influence on the regions for which a response is consistent, i.e., eastern England, is small. Therefore the winter NAO might only explain a small portion of the observed shift in eastern England.

To explain the potential role of the summer NAO and the East Atlantic pattern for the consistent shift from summer to autumn peak times further research needs to be carried out. For monthly mean precipitation both the NAO and the East Atlantic pattern have already been shown to have a significant influence on summer precipitation throughout the British Isles (Murphy and Washington 2001, Bladé et al 2012b).

The changes in the relative amplitude of heavy precipitation for the middle and end of the 21st century for the western regions is coherent with already observed changes during the end of the 20th century (Fowler and Kilsby 2003b). The inter-AOGCM variability of the relative amplitude's response is larger than the inter-RCM variability⁷. ECHAM5-driven simulations show a positive trend, BCM/ARPEGE-driven simulations show a rather negative trend and HadCM3-driven simulations do not show a common signal. Thus either the relative amplitude is dominated by internal climate variability or at least two of the three AOGCMs do not correctly simulate the sign of future trends. To identify the contribution of internal variability and of uncertainties from driving AOGCMs, initial condition ensembles and a wider range of driving AOGCMs are needed.

The mechanisms describing the relative amplitude as well as the peak times of heavy precipitation need further investigation. For instance, it is not clear to what extent the Scandinavian, East Atlantic and other teleconnection patterns are of relevance for the relative amplitude of extremes. Both the Scandinavian pattern's and the East Atlantic pattern's representation in the CMIP3 AOGCMs vary considerably between climate models (Handorf and Dethloff 2009). This could be a starting point for explaining the dependence of the relative amplitude's change on the driving AOGCM.

For readability we sketch the statistical model described by Schindler et al (2012). We model rare events as an inhomogeneous Poisson process which is characterized by its intensity measure on [0,1] × [z,∞], for z greater than a suitable threshold u:

$$\begin{eqnarray} &&\left [1+\xi \left (\frac{z-\mu }{\sigma }\right )\right ]^{-1/\xi }, \end{eqnarray} \tag{ A.1 }$$

where μ,σ and ξ are called location, scale and shape parameters. The intensity measure describes the probability of exceeding a level z.

These parameters are adjusted such that at each time t the annual cycle is accounted for:

$$\begin{eqnarray} &&\fl \mu (t)={\mu }_{0}+{\mu }_{1}\sin \left (\frac{2 \pi t}{3 6 5.2 5}\right )+{\mu }_{2}\cos \left (\frac{2 \pi t}{3 6 5.2 5}\right ), \end{eqnarray} \tag{ A.2 }$$

$$\begin{eqnarray} &&\fl \sigma (t)={\sigma }_{0}+{\sigma }_{1}\sin \left (\frac{2 \pi t}{3 6 5.2 5}\right )+{\sigma }_{2}\cos \left (\frac{2 \pi t}{3 6 5.2 5}\right ). \end{eqnarray} \tag{ A.3 }$$

Since the shape parameter is difficult to estimate with limited amount of data, it is kept constant (Rust et al 2009). Introducing additional (seasonally varying) parameters would lead to very uncertain estimates. For the realizations z_i of the random variable Z, for i = 1,...,N_obs, where N_obs is the number of observations, one can estimate the parameters, (μ₀,μ₁,μ₂,σ₀,σ₁,σ₂,ξ)≕θ, by maximizing the likelihood function with respect to θ. In general it is easier to minimize the negative log-likelihood function than to maximize the likelihood function. With δ_i = 1 if z_i > u_i and δ_i = 0 else we hence obtain

$$\begin{eqnarray*} &&\fl {l}_{n}(\theta )=-\frac{1}{{n}_{p}}\sum _{i=1}^{{N}_{\mathrm{obs}}}\left (1+\xi \frac{{u}_{i}-{\mu }_{i}}{{\sigma }_{i}}\right )^{-1/\xi }\nonumber\\ &&+~\sum _{i=1}^{{N}_{\mathrm{obs}}}{\delta }_{i}\log \left (\frac{1}{{\sigma }_{i}}\left (1+\xi \frac{{z}_{i}-{\mu }_{i}}{{\sigma }_{i}}\right )^{(-1/\xi -1)}\right ) \end{eqnarray*}$$

with n_p being the number of observations per block and the non-stationarity of the parameters denoted as μ_i:=μ(t_i) and σ_i:=σ(t_i), t_i∈{1,...,365.25 × 4} being the time of the observation z_i in the 4 yr cycle.

To select rare events a high threshold u ≫ 1 is fixed, such that above it, the asymptotic marginal and dependence properties appear to be stable (Davison and Smith 1990). We select u by choosing the 95%-quantile of the anomalies with respect to the climatological mean over the 40 yr period as a threshold. We apply an automatic procedure to check the goodness-of-fit: we simulate a 95% confidence interval (CI) for the quantile–quantile plot and count the points outside the CI. If more than 5% of the threshold exceedances lie outside the CI, the fit is considered critical. If 5% or less of the grid points of one data set indicate a critical fit, this is considered statistically expected and the threshold is not changed. If there are more than 5% critical grid points in one data set, we increase the threshold by one per cent, refit the statistical model and start over again. The iteration halts if the threshold reaches the 97%-quantile, the remaining critical grid points are excluded from the subsequent analysis.

The annual cycle of the monthly return levels is defined via the twelve monthly return levels. In this non-stationary setting the calculation of the return levels, z_p, is an inverse problem (Coles 2001):

$$\begin{eqnarray} &&1-\frac{1}{p}=\Pr(\max ({X}_{1},\ldots ,{X}_{n})\leq {z}_{p})\approx \mathop{\prod }\limits _{j\in \{ 1,\ldots ,{n}_{p}\} }{p}_{j} \end{eqnarray} \tag{ A.4 }$$

where

$$\begin{eqnarray*} &&{p}_{j}=\left \{ \begin{array}{@{}l@{}} \displaystyle 1-{n}_{p}^{-1}(1+\xi ({z}_{p}-{\mu }_{j})/{\sigma }_{j})^{-1/\xi },\\ \displaystyle \tqs \quad \mathrm{if}~1+\xi ({z}_{p}-{\mu }_{j})/{\sigma }_{j}\geq 0\\ \displaystyle 1,\tqs \mathrm{otherwise} \end{array}\right. \end{eqnarray*}$$

with the non-stationarity of the parameters denoted as μ_j:=μ(t_j) and σ_j:=σ(t_j) for j∈{1,...,n_p} and n_p being the number of observations per block. To calculate the annual return level z_p the block length is set to 365, i.e., n_p = 365. To calculate monthly return levels, z_p(m), (non-stationary setting) the product over the index set is restricted to a specific month, m. This can be achieved by choosing j∈{1,...,n_p} in (A.4) such that t_j∈m (January: j∈{1,...,31}; February: j∈{32,...,60}; etc).